Yale University
Department of Computer Science
Cryo-EM Structure Determination through
Eigenvectors of Sparse Matrices
Ronald. R. Coifman Yoel ShkolniskyFred J. Sigworth Amit Singer
YALEU/DCS/TR-1389
Cryo-EM Structure Determination through
Eigenvectors of Sparse Matrices
Ronald. R. Coifman∗, Yoel Shkolnisky∗, Fred J. Sigworth†
and Amit Singer∗
Abstract
Recovering the three-dimensional structure of proteins is impor-
tant for understanding their functionality. We describe a spectral
graph algorithm for reconstructing the three-dimensional structure of
molecules from their cryo-electron microscopy images taken at random
unknown orientations. The key idea of the algorithm is designing a
sparse operator defined on the projection images, whose eigenvectors
reveal their orientations.
The special geometry of the problem rendered by the Fourier projection-
slice theorem is incorporated into the construction of a weighted graph
whose vertices are the radial Fourier lines and whose edges are linked
with the common line property. The radial lines are associated with
points on the sphere and are networked through spider like connec-
tions. The graph organizes the radial lines on the sphere in a global
manner that reveals the projection directions. This organization is de-
rived from a global computation of a few eigenvectors of the graph’s
adjacency matrix. Once the directions are obtained, the molecule can
be reconstructed using classical tomography methods.
∗Department of Mathematics, Program in Applied Mathematics, Yale Univer-sity, 10 Hillhouse Ave. PO Box 208283, New Haven, CT 06520-8283 USA.{[email protected], [email protected], [email protected]}
†Department of Cellular and Molecular Physiology, Yale University School of Medicine,333 Cedar Street, New Haven, CT 06520 USA. [email protected]
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The presented algorithm is direct (as opposed to iterative refine-
ment schemes), does not require any prior model for the reconstructed
object, and shown to have favorable computational and numerical
properties. Moreover, the algorithm does not impose any assump-
tion on the distribution of the projection orientations. Physically, this
means that the algorithm successfully reconstructs molecules that have
unknown spatial preference.
We also introduce extensions of the algorithm, based on the spec-
tral properties of the operator, which significantly improve its ap-
plicability to realistic data sets. These extensions include: particle
selection, to filter corrupted projections; center determination to es-
timate the relative shift of each projection; and, a method to resolve
the heterogeneity problem, in cases where a mix of different molecules
is being imaged concurrently.
1 Introduction
“Three dimensional electron microscopy” [1] is the name commonly given
to methods in which the 3D structures of macromolecular complexes are
obtained from sets of images taken by an electron microscope. The most
widespread and general of these methods is single-particle reconstruction
(SPR). In SPR the 3D structure is determined from images of randomly
oriented and positioned identical macromolecular “particles”, typically com-
plexes 500 kDa or larger in size. The SPR method has been applied to
images of negatively stained specimens, and to images obtained from frozen-
hydrated, unstained specimens [2]. In the latter technique, called cryo-
electron microscopy (cryo-EM) the sample of macromolecules is rapidly frozen
in a thin (∼ 100 nm) layer of vitreous ice, and maintained at liquid nitrogen
temperature throughout the imaging process.
SPR from cryo-EM images is of particular interest because it promises
to be an entirely general technique. It does not require crystallization or
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other special preparation of the complexes to be imaged, and in the future
it is likely to reach sufficient resolution (∼ 0.4 nm) to allow the polypeptide
chain to be traced and residues identified in protein molecules [3]. Even
at the present best resolutions of 0.9–0.6 nm, many important features of
protein molecules can be determined [4].
Much progress has been made in algorithms that, given a starting 3D
structure, are able to refine that structure on the basis of a set of negative-
stain or cryo-EM images, which are taken to be projections of the 3D object.
Data sets typically range from 104 to 105 particle images, and refinements
require tens to thousands of CPU-hours. As the starting point for the refine-
ment process, however, some sort of ab initio estimate of the 3D structure
must be made. Present algorithms are based on the “Angular Reconstitu-
tion” method of van Heel [5] in which a coordinate system is established from
three projections, and the orientation of the particle giving rise to each image
is deduced from common lines among the images.
A serious limitation and an active area of research in SPR image process-
ing is the problem of heterogeneity [6, 7]. Often a set of images is obtained
from a mixture of particles of two or more different kinds or different confor-
mations. Because traditional SPR methods assume identical particles, they
fail to distinguish the different particle species. Extremely desirable would
be the ability to establish two or more ab initio reconstructions from a single
set of images.
We propose Globally Consistent Angular Reconstitution (GCAR), a recon-
struction algorithm that does not assume any ab initio model and establishes
a globally consistent coordinate system from all projections. The special ge-
ometry of the problem rendered by the Fourier projection-slice theorem [8]
is incorporated by GCAR into a weighted directed graph whose vertices are
the radial Fourier lines and whose edges are linked using the common line
property. Radial lines are viewed as points on the sphere and are networked
through spider-like connections. The graph organizes the radial lines on the
4
sphere in a global manner and reveals the projection directions. Such an
organization is derived from a global computation of the eigenvectors of the
sparse graph matrix. This global averaging property makes GCAR robust
to both noise and false detections of common lines. The global organization
of the common lines also promises to provide a solution to the problems of
center determination and heterogeneity. GCAR is extremely fast because it
requires only the computation of a few eigenvectors of a sparse matrix.
Once the directions are revealed by the eigenvectors, the reconstruction is
performed using the combination of classical tomography methods together
with more recent non-uniform FFT techniques [9, 10].
We have successfully applied similar graph-based approaches to recon-
struct two-dimensional structures, such as the Shepp-Logan phantom, from
noisy one-dimensional projections taken at random directions [11]. Many of
the recent and successful algorithms for nonlinear dimensionality reduction of
high-dimensional data, such as locally linear embedding (LLE) [12], Hessian
LLE [13], Laplacian eigenmap [14] and diffusion maps [15] involve the com-
putation of eigenvectors of data-dependent sparse kernel matrices. However,
such algorithms fail to solve the cryo-EM problem, because the reduced co-
ordinate system that each of them obtains does not agree with the projection
directions. On the other hand, GCAR finds the desired coordinate system
of projection images, because it is tailored to the geometry of the problem
through the Fourier projection-slice theorem.
The organization of this report is as follows. In Section 2 we introduce the
cryo-electron microscopy problem, as well as the related mathematical back-
ground. We introduce the GCAR operator, which reveals the orientations of
the projections, and analyze its properties in Section 3. The algorithm for
recovering the projection orientations is summarized in Section 4, together
with a few technical implementation details. Examples of applying this algo-
rithm to two phantoms are given in Section 5. In Section 6 we present several
useful extensions of the algorithm. In particular, we present how to deter-
5
mine the center of each projection by utilizing the common-line property in
Section 6.1, how to solve the heterogeneity problem, where projections of two
types of molecules are mixed together, in Section 6.2, and how to exploit the
properties of the GCAR graph to filter out “bad” projections in Section 6.3.
We conclude in Section 7 with a summary and a description of future work.
2 Problem Setup
2.1 The physical setting
The cryo-EM reconstruction problem is to find the three-dimensional struc-
ture of a molecule given samples of its two-dimensional projection images at
unknown random directions. The intensity of pixels in a given projection
image corresponds to line integrals of the electric potential induced by the
molecule along the path of the electrons. The highly intense electron beam
destroys the molecule and it is therefore impractical to take projection im-
ages of the same molecule at known different directions, as in the case of
classical computerized tomography (CT). In other words, a single molecule
can be imaged only once. All molecules are assumed to have the exact same
structure; they differ only by their spatial orientation. Thus, every image is
a projection of the same molecule but at an unknown random orientation.
The locations of the microscope (source) and the camera/film (detectors)
are fixed, while different images correspond to different spatial rotations of
the molecule. Every image is thus associated with an element of the rotation
group SO(3). If the electric potential of the molecule is φ(r) in some fixed
‘laboratory’ coordinate system r = (x, y, z), then rotating the molecule by
g ∈ SO(3) results in the potential φg(r) = φ(g−1r). The projection image
Pg(x, y) is obtained by integrating φg(r) along the z-direction (the source-
detector direction)
Pg(x, y) =
∫ ∞
−∞
φg(x, y, z) dz.
6
Projection
Molecule
Electronsource
g∈SO(3)
Figure 2.1: Schematic drawing of projection.
The projection operator is also known as the X-ray transform [8]. Figure 2.1
is a schematic illustration of the cryo-EM setting.
The projection image is a digital picture given as an p× p grid of pixels
Pg = {Pg(xi, yj)}pi,j=1 ∈ R
p2 , where p is determined by the characteristics of
the imaging setup.
The cryo-EM problem is stated as follows: find φ(x, y, z) given a collection
of K projections {Pgk}Kk=1, where gk are unknown rotations. If the rotations
{gk}Kk=1 were known then the reconstruction of φ(x, y, z) could be performed
by classical tomography methods. Therefore, the cryo-EM problem is re-
duced to estimating the rotations {gk}Kk=1 given the data set {Pgk}
Kk=1.
For convenience, we adopt the following equivalent point of view. Instead
of having the microscope fixed and the molecule oriented randomly in space,
we think of the molecule as being fixed, and the microscope being the one that
is randomly rotated in space. The orientation of the microscope is described
by the beaming direction θ ∈ S2 (axis of rotation) and the in-plane rotation
7
angle α ∈ [0, 2π) of the camera.
2.2 Fourier projection-slice theorem
The two-dimensional Fourier transform of a projection image P (x, y) is given
by the double integral
P̂ (ω) =1
(2π)2
∫
θ⊥e−ir·ωP (r) dr, (1)
where θ is the projection direction, that is, a normal to the image plane. The
three-dimensional Fourier transform of the molecule is given by the triple
integral
φ̂(ξ) =1
(2π)3
∫
R3
e−ir·ξφ(r) dr. (2)
One of the cornerstones of tomography is the Fourier projection-slice theorem
stating that the two-dimensional Fourier transform of a projection image is
a planar slice θ⊥ of the three-dimensional Fourier transform of the molecule
(see, e.g., [8, p. 11])
P̂ (η) = 2πφ̂(η), η ∈ θ⊥. (3)
Figure 2.2 depicts the geometry induced by the Fourier projection-slice the-
orem. The sphere represents the 3D Fourier space of the molecule, and the
planes are three particular two-dimensional slices of it. As the 2D Fourier
transform of a projection image corresponds to a slice of the 3D Fourier
space, any two slices share a common line, i.e., the intersection line of the
two planes. Figure 2.2 shows three particular slices that happen to share the
same common line, marked in dashed black. Note that the effect of rotating
the camera (changing α) while fixing the beaming direction θ is an in-plane
rotation of the slice without changing its position.
The Fourier transform of each projection image can be considered in polar
coordinates as a set of planar radial lines in two-dimensional Fourier space.
As a result of the Fourier projection-slice theorem, every planar radial line
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Figure 2.2: Two-dimensional slices of the three-dimensional Fourier space.
is also a radial line in the three-dimensional frequency ball of Fig. 2.2, and
there is a one-to-one correspondence between those Fourier radial lines and
points on the unit sphere S2. The radial lines of a single projection image
correspond to a great circle (a geodesic circle) on S2. For example, the radial
lines of the green slice in Fig. 2.2 form the equator. Thus, to every projection
image Pgk there corresponds a unique great circle Ck over S2, and the common
line property is restated as follows: any two different geodesic circles over S2
intersect at exactly two antipodal points.
If the projection directions (the gk’s or the (θk, αk)’s) were known, then
the 2D polar Fourier transform of the projection images would have resulted
in the values of φ̂(ξ) on different 3D radial lines, as stated by Eq. 3. Fourier
inversion of φ̂(ξ) in a frequency ball of radius B (|ξ| < B, where B is de-
termined by the sampling resolution) would have then reconstructed a band
limited approximation of φ. However, in the cryo-EM problem the slices are
unorganized. Neither their directions θ nor their in-plane rotations α are
known.
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2.3 Correspondence between Fourier rays and S2
We next explain the discretization of Fourier space and derive a mapping
between the discretized Fourier space and points on the unit sphere S2. Such
a mapping would allow us to proceed by exploiting the simple geometry of
S2.
Let P1(x, y), . . . , PK(x, y) be K projection images. Upon writing the
Fourier transform in Eq. 1 in polar coordinates, we obtain
P̂k(ρ, ω) =1
(2π)2
∫ ∫
Pk(x, y)e−i(xρ cos ω+yρ sinω) dx dy, k = 1, . . . , K. (4)
For digital implementations we discretize ρ and ω, and compute Eq. 4 using a
nonequally-spaced FFT [9]. We denote by L the angular discretization of ω,
and sample ρ in n equally-spaced points. That is, we split each transformed
projection into L radial lines Λk,0, . . . ,Λk,L−1 ∈ Rn, each represented by a
set of n equispaced points
Λk,l =(
P̂k(B/n, 2πl/L), P̂k(2B/n, 2πl/L), . . . , P̂k(B, 2πl/L))
∈ Rn, (5)
for 1 ≤ k ≤ K, 0 ≤ l ≤ L− 1, where B is the band limit. Note that the DC
term (ρ = 0 in Eq. 4) is shared by all lines independently of the image and
can therefore be ignored.
Let
Λ(β) =(
2πφ̂(βB/n), 2πφ̂(2βB/n), . . . , 2πφ̂(nβB/n))
∈ Rn (6)
be n samples from a ray through the origin in 3D Fourier space, in a direction
given by the unit vector β ∈ S2. This defines a map Λ : S2 → Rn that maps
each unit vector in R3 to n samples of the Fourier ray in that direction.
According to the Fourier projection-slice theorem, the radial lines Λk,l are
the evaluations of the function Λ(β) at the unknown points βk,l ∈ S2. The
function Λ(β) is unknown, because so is φ̂. Our goal is to find the sources
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βk,l of the overall KL radial lines. The L sources {βk,l}L−1l=0 (k fixed) are
equidistant points on the great circle Ck ⊂ S2.
3 Orientation Revealing Operator
In this section we introduce the GCAR operator, whose eigenvectors reveal
the orientation of each projection. The GCAR operator is a graph with every
node representing a ray in Fourier space, and whose edges are determined by
the common-line property. The formal construction of this graph is presented
in Section 3.1. The normalized adjacency matrix of the graph can be viewed
as an averaging operator for functions defined on the nodes of the graph,
as explained in Section 3.2. Analyzing the eigenvectors of this operator in
Section 3.3 shows that they encode the projection orientations. We conclude
by showing in Section 3.4 that the eigenvectors of the GCAR matrix are
intimately related to the spherical harmonics.
3.1 GCAR graph
We assume as before that we are given K projection images. Let Λk,l, k =
1, . . . , K, l = 1, . . . , L be KL Fourier rays computed from the K projection
images. We think of the radial lines as vertices of a graph, where each radial
line Λk,l and its source βk,l are identified with the vertex indexed by the pair
(k, l). In other words, the set of vertices V of the directed graph G = (V,E)
is
V = {(k, l) : 1 ≤ k ≤ K, 0 ≤ l ≤ L− 1},
where the number of vertices is |V | = KL. The set E ⊂ V × V of directed
edges (or arrows)
E = {((k1, l1), (k2, l2)) : (k1, l1) points to (k2, l2)}
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will be defined shortly. The graphG is represented using a (sparse) adjacency
matrix W of size KL×KL
W(k1,l1),(k2,l2) =
{
1 if ((k1, l1), (k2, l2)) ∈ E
0 if ((k1, l1), (k2, l2)) 6∈ E.
Each row of W corresponds to one radial line in Fourier space Λk,l, or al-
ternatively to a point βk,l on S2. The nonzero elements in that row, when
viewed on S2, constitute a spider-like neighborhood, as will be explained
shortly.
We now define the set E. Consider a specific vertex (k1, l1), that is, the
radial line Λk1,l1. Identify its 2J + 1 neighboring radial lines from its own
transformed projection P̂gk1 (indicated in red in the top panel of Fig. 3.1)
and set
((k1, l1), (k1, l1 + l)) ∈ E for − J ≤ l ≤ J, (7)
where addition is modulo L. We link every radial line with J neighboring
radial lines in each direction, clockwise and counterclockwise.
We next determine the remaining nonzero entries in row (k1, l1) of W by
using the common-line property. Whenever we identify Λk1,l1 ≈ Λk2,l2 as the
common line between projection images k1 and k2, we add to E the following
links
((k1, l1), (k2, l2 + l)) ∈ E for − J ≤ l ≤ J. (8)
The antipodal rays Λk1,l1+L/2 and Λk2,l2+L/2 are common radial lines as well
and are linked together in the same manner by setting
((k1, l1 + L/2), (k2, l2 + L/2 + l)) ∈ E for − J ≤ l ≤ J. (9)
In general, the adjacency matrix W is nonsymmetric: ((k1, l1), (k2, l2)) ∈
E (for k1 6= k2) reflects the fact that the circle Ck2 containing (k2, l2) passes
nearby the point (k1, l1). However, Ck1 does not necessarily passes nearby
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Projection Fourier transform 3D Fourierspace
3D Fourierspace
Fourier transformProjection
Pgk1 P̂gk1
(k1,l1)
Λk2,l2=Λk1,l1Λk2,l2=Λk1,l1
Pgk2 P̂gk2
Figure 3.1: Fourier projection-slice theorem.
(k2, l2) so ((k2, l2), (k1, l1)) 6∈ E. Symmetry occurs only within the same
circle, that is, ((k, l1), (k, l2)) ∈ E ⇐⇒ ((k, l2), (k, l1)) ∈ E, which happens
whenever |l1 − l2| ≤ J . Although the size of W is KL × KL, by choosing
J ≪ L we force it to be a sparse matrix. Its number of nonzero entries is
only
|E| = (2J + 1)KL+ 2K(K − 1)(2J + 1). (10)
The first summand corresponds to the 2J + 1 edges in Eq. 7 for each of the
KL vertices. The second summand corresponds to edges between different
images (Eq. 8). Any two circles intersect at exactly two antipodal points,
so there are 2(
K2
)
= K(K − 1) meeting points. Every meeting point, that
is, every common-line Λk1,l1 = Λk2,l2 contributes 2J + 1 nonzero elements in
row (k1, l1) of W , and 2J + 1 nonzero elements in row (k2, l2). In practice,
the number of edges in E is smaller than the number in Eq. 10, as explained
in Section 4.
If we take row (k1, l1) from W and plot the points βk,l ∈ S2 for which
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(a) One row of W – one spider (b) Two rows of W – two spiders
Figure 3.2: Mapping the nonzero entries of W to S2.
W (k1,l1),(k,l) = 1, we get a picture that looks like Fig. 3.2a. In light of
Fig. 3.2a, we refer to each row of W as the “spider that corresponds to the
point (k1, l1)”. The point βk1,l1 is the head (center) of the spider. Sources
βk,l that correspond to the same k (come from the same projection image)
are marked with the same color. Figure 3.2b shows two spiders on S2 for a
certain randomized choice of circles with K = 200, L = 100, J = 10, from
which we can see how different spiders interact. This interaction is essential
for the global consistent assignment of coordinates explained below.
Different spiders may have different number of legs, so row sums of W
may be different. The outdegree dk,l of the (k, l)’th vertex is the sum of the
corresponding row in W
dk,l =∑
(k′,l′)∈V
W(k,l),(k′,l′) = |{(k′, l′) : ((k, l), (k′, l′)) ∈ E}|. (11)
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3.2 Averaging operator
We normalize the adjacency matrixW to have constant row sums by dividing
it by a diagonal matrix D whose diagonal elements equal the outdegrees dk,l
of the vertices (or equivalently the row sums of W )
D(k,l),(k,l) = dk,l, (12)
with dk,l given by Eq. 11. This normalization results in the operator
A = D−1W . (13)
The operator A : C|V | → C|V | takes any discrete complex valued function
f : V → C (realized as a vector in CKL) and assigns the head of each spider
the average of f over the entire spider
(Af )(k1, l1) =1
dk1,l1
∑
((k1,l1),(k2,l2))∈E
f(k2, l2).
We therefore regard A as an averaging operator over C|V |.
The matrix A is row stochastic (the row sums of A all equal 1), and
therefore the constant function ψ0 = 1 (ψ0(v) = 1, ∀v ∈ V ) is an eigenvector
with λ0 = 1: Aψ0 = ψ0. The remaining eigenvectors may be complex
and come in conjugate pairs, because A is real but not symmetric: Aψ =
λψ ⇐⇒ Aψ̄ = λ̄ψ̄. As of the spectrum of A, λ0 = 1 is the largest possible
eigenvalue and the remaining eigenvalues reside inside the complex unit disk
|λ| < 1.
3.3 Coordinate eigenvectors
The operator A has many interesting properties. For the cryo-EM problem,
the most important property is that the coordinates of the sources βk,l are
eigenvectors of the averaging operator A, sharing the same eigenvalue. Ex-
15
plicitly, let x, y, z : R3 → R be the coordinate functions in R3. Then, the
vectors x,y, z ∈ RKL defined by
x = x(βk,l), y = y(βk,l), z = z(βk,l),
k = 1, . . . , K, l = 1, . . . , L are eigenvectors of A.
This remarkable fact is a consequence of the following observation: the
center of mass of every spider is in the direction of the spider’s head, because
any pair of opposite legs balance each other. For example, the center of mass
of a spider whose head is located at the north pole lies just beneath it.
We include the details of the rather technical proof. Suppose f(x, y, z) =
a1x + a2y + a3z = a · β is a linear function, where a = (a1, a2, a3) and
β = (x, y, z) ∈ S2. Consider a spider whose head is at the point β1 =
(x1, y1, z1) ∈ S2, where the value of the function is f(x1, y1, z1) = a · β1.
Let β2,β3 be two unit vectors that complete β1 into an orthonormal system
of R3. In other words, the 3 × 3 matrix U whose columns are β1,β2,β3
is orthogonal. We express any point β = (x, y, z) on the sphere as a linear
combination β = x′β1+y′β2+z
′β3 = Uβ′, where β′ = (x′, y′, z′) is a rotated
coordinate system. We apply a change of variable β → β′ in f to obtain
the linear function f ′(x′, y′, z′) = f(x, y, z) = a · β = a · Uβ′ = a′ · β′,
where a′ = UTa = (a′1, a′2, a
′3). The parameterization of a great circle going
through β1 is
cos θβ1 + sin θ cosϕ0β2 + sin θ sinϕ0β3,
where θ ∈ (−π, π] and ϕ0 is a fixed parameter that determines the direction
of the circle. On that circle, f is a function of the single parameter θ
f(θ) = f ′(cos θ, sin θ cosϕ0, sin θ sinϕ0) = a′ · (cos θ, sin θ cosϕ0, sin θ sinϕ0).
16
The average f̄ of f over the two discrete opposite legs of that circle is
f̄(x1, y1, z1) =1
2J + 1
J∑
l=−J
f
(
2πl
L
)
=a′
2J + 1·
J∑
l=−J
(cos2πl
L, sin
2πl
Lcosϕ0, sin
2πl
Lsinϕ0)
=
[
1
2J + 1
J∑
l=−J
cos2πl
L
]
a′ · (1, 0, 0),
due to the linearity of the dot product and the fact that sin θ is an odd
function. From
a′ · (1, 0, 0) = UTa · (1, 0, 0) = a ·U(1, 0, 0) = a · β1 = f(x1, y1, z1),
we conclude that
f̄(x1, y1, z1) =
[
1
2J + 1
J∑
l=−J
cos2πl
L
]
f(x1, y1, z1) (14)
holds for all (x1, y1, z1) and for any circle going through it. Therefore, linear
functions are eigenvectors of the averaging operator A with eigenvalue λ =1
2J+1
∑Jl=−J cos
2πlL
. This completes the proof.
3.4 Spherical harmonics
The eigenfunctions of the Laplacian operator on the sphere S2 are known to
be the spherical harmonics Y ml [8, p.195](also known as the eigenstates of
the angular momentum operator in quantum mechanics)
∆S2Yml = −l(l + 1)Y
ml , l = 0, 1, 2, . . . , m = −l,−l + 1, . . . , l. (15)
17
The (non-normalized) spherical harmonics are given in terms of the associ-
ated Legendre polynomials of the zenith angle θ ∈ [0, π] and trigonometric
polynomials of the azimuthal angle ϕ ∈ [0, 2π)
Y 0l (θ, ϕ) = Pl(cos θ),
Y ml (θ, ϕ) = P|m|l (cos θ) cosmϕ, 1 ≤ m ≤ l,
Y −ml (θ, ϕ) = P|m|l (cos θ) sinmϕ, 1 ≤ m ≤ l,
while the Laplacian is given by
∆S2 =1
sin θ
∂
∂θ
(
sin θ∂
∂θ
)
+1
sin2 θ
∂2
∂ϕ2. (16)
The eigenspaces are degenerated so that the eigenvalue l(l + 1) has multi-
plicity 2l+1. Alternatively, the l’th eigenspace corresponds to homogeneous
polynomials of degree l restricted to S2. In particular, the first three non-
trivial spherical harmonics Y m1 share the same eigenvalue and are given by
the three linear functions
Y 11 = x, Y−11 = y, Y
01 = z.
The spherical harmonics Y ml are usually derived by separating variables
in Eqs. 15–16. However, the fundamental reason for which the spherical har-
monics are eigenfunctions of the Laplacian is that the latter commutes with
rotations. In fact, the classical Funk-Hecke theorem (see, e.g., [8, p. 195])
asserts that the spherical harmonics are the eigenfunctions of any integral op-
erator K : L2(S2) → L2(S2) that commutes with rotations. Such operators
are of the form
(Kf)(β) =
∫
S2k(〈β,β′〉)f(β′) dSβ′,
where k : [−1, 1] → R is the kernel function that depends only on the angle
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between β,β′ ∈ S2. For such integral operators we have
KY ml = λlYml ,
where the eigenvalues λl depend on the specific kernel function k(·) and are
given by
λl = 2π
∫ 1
−1
k(t)Pl(t) dt.
For example, the spherical harmonics are the eigenfunctions of the operator
that corresponds to averaging over spherical caps.
The averaging operator A defined in Section 3.2 does not commute with
rotations, because every spider has different number of legs that go in dif-
ferent directions. The averaging operator commutes with rotations only in
the limit of infinite number of projection images corresponding to a uniform
distribution over SO(3) (the Haar measure). Although A does not commute
with rotations and the Funk-Hecke theorem does not hold, the coordinate
vectors x,y, z ∈ RKL span an eigenspace of A, due to the center of mass
property. Figure 3.3 depicts the first 50 eigenvalues of the operator A con-
structed using K = 200 random points on S2, L = 100 points on each
geodesic circle, and J = 10 samples on each leg of the spider. This corre-
sponds to using K = 200 projection images, L = 100 radial Fourier lines for
each projection, and using J = 10 in Eqs. 7–9. The threefold multiplicity
corresponding to the coordinate vectors is clearly seen in Fig. 3.3. Moreover,
the observed numerical multiplicities of 1, 3, 5, 7 and 9 are explained by the
spherical harmonics. The remaining eigenvalues seen in Fig. 3.3 correspond
to clustering of the circles. By clustering we mean that each of the corre-
sponding eigenvectors takes an almost constant value on one of the circles
and practically vanishes on all other circles.
19
0 5 10 15 20 25 30 35 40 450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.3: Numerical spectra of A: K = 200, L = 100, J = 10.
4 Algorithm
The fact that the coordinates x,y, z of the sources βk,l, k = 1, . . . , K,
l = 1, . . . , L, form an eigenspace of A (see Section 3.2) enables to com-
pute them by computing the first three non-trivial eigenvectors ψ1,ψ2,ψ3
of the sparse matrix A. Taking a sufficiently small J ensures that x,y, z
appear immediately after ψ0 = 1 in the spectrum of A. However, due to
the threefold multiplicity of the eigenvalue, the computed eigenvectors may
be any linear combination of the coordinate vectors. This linear combina-
tion is easily determined (up to an orthogonal transformation) by using the
fact that the coordinates must correspond to points on the sphere (i.e., unit
length vectors). To this end, we look for a 3 × 3 matrix M such that
X ≡
− xT −
− yT −
− zT −
= M
− ψT1 −
− ψT2 −
− ψT3 −
≡MΨ. (17)
The diagonal of the KL×KL matrix XTX = ΨTMTMΨ is all ones, be-
cause the points are on the unit sphere, that is, ‖βk,l‖2 = x2(k, l)+y2(k, l)+
z2(k, l) = 1. We end up with an overdetermined system of KL linear equa-
20
tions(
ΨTMTMΨ)
ii= 1, (18)
for the 9 entries of MTM . The least squares solution for MTM is then
followed by an SVD or a Cholesky decomposition to yield M . We can
recover M only up to an orthogonal transformation O ∈ O(3), because
MTOTOM = MTM . The reconstruction of the molecule is up to an arbi-
trary rotation and possibly a reflection (the chirality or handedness cannot
be determined).
The locations of the radial lines can be further refined by using the fact
that same image radial lines correspond to a great circle on S2. In particular,
such radial lines belong to the same plane (slice). Therefore, we improve the
estimation of coordinates by using principal component analysis (PCA) for
groups of L radial lines at a time. Furthermore, we equally space those radial
lines on the corresponding great circle.
GCAR is summarized in Algorithm 1. Step 3 of finding pairs of common
lines can be done efficiently in linear time complexity (instead of comparing
a quadratic number of pairs) by approximate nearest neighbors algorithms in
a low dimensional feature space. Note that the algorithm assumes that the
projection images are centered, for otherwise an arbitrary phase, which alters
the step of finding common lines, comes into play. Handling non-centered
projections is addressed in Section 6.1.
The eigenvector computation is global and takes into account all the
local pieces of information about common lines. Even if some common lines
are misidentified, those errors are averaged out in the global eigenvector
computation. Thus, GCAR should be regarded as a very efficient way of
integrating the local cryo-EM geometry into a global orientation assignment.
The construction of the matrix W , as described in Section 3.1, uses all
pairs of common lines. That is, for each pair of projection images k1 and k2,
we find the Fourier lines Λk1,l1 and Λk2,l2 such that Λk2,l2 is closest to Λk1,l1 ,
and use the pair (k1, l1) and (k2, l2) to add edges to the set E in Eqs. 8–9.
21
As can be seen in Fig. 2.2, this corresponds to finding all geodesic circles
on S2 that pass though βk1,l1. Note however, that the coordinates vectors
are eigenvectors of A = D−1W even if we use only a few of the geodesic
circles that go through βk1,l1 . This corresponds to using fewer legs in each
spider. Moreover, the resulting matrix W is sparser, and so requires less
memory and its eigenvectors can be computed faster. The key advantage
of this observation is that we do not need to use all lines determined by
the(
K2
)
intersection of projection images. We use only pairs of images for
which Λk1,l1 is very close to Λk2,l2 . This results in fewer misidentifications of
common-lines, and leads to a more accurate estimation of the orientations.
This is demonstrated in Section 5.
Algorithm 1 Outline of GCAR
Require: Projection images Pk(x, y), k = 1, 2, . . . , K1: Compute the polar Fourier transform P̂k(ρ, ω) (Eq. 4).2: Split each P̂k(ρ, ω) into L radial lines Λk,l (Eq. 5).3: Find common lines Λk1,l1 ≈ Λk2,l2.4: Construct the sparse KL×KL weight matrix W with J ≪ L (following
Section 3.1).5: Normalize W by its outdegree D and form the averaging operator A =D−1W (Eq. 13).
6: Compute the first three non-trivial eigenvectors of A: Aψ1 =λψ1, Aψ2 = λψ2, Aψ3 = λψ3.
7: Unmix x,y, z from ψ1,ψ2,ψ3.8: Refinement: PCA and equally space same image radial lines.
5 Numerical Examples
The algorithm was implemented in MATLAB and was tested on two types of
phantoms. The first phantom consists of a set of ellipsoids, whose projections
can be computed analytically. This phantom is shown in Fig. 5.1a. The
second phantom is a 96 × 96 × 96 density map of the E. coli ribosome,
22
whose projections were computed by approximating line integrals through
the volume. The E. coli phantom is pictured in Fig. 5.5a. All tests used
K = 100 projection images, with L = 300 radial Fourier lines per projection,
and 100 samples along each Fourier ray. The projections of the analytic
phantom were of size 97 × 97 to avoid half pixel shifts involved in even-size
projections. The projections of the E. Coli phantom were of size 96 × 96
and center estimation was used to compensate for half pixel shifts. The
polar Fourier transform was computed as described in [16]. Common-lines
between two Fourier-transformed projections were found by computing and
comparing correlations between all Fourier lines of the two projections. No
knowledge of the orientations or their distribution was used by the algorithm.
All tests were executed on a quad core Xeon 2.33GHz running Linux.
Once orientations were determined, the phantom was reconstructed by inter-
polating theKL Fourier lines into the three-dimensional pseudo-polar grid by
using nearest-neighbor interpolation followed by an inverse 3D pseudo-polar
Fourier transform, implemented along the lines of [10, 17].
Figure 5.1a shows a 3D rendering of the analytic phantom, with several of
its projections given in Fig. 5.2. The projection orientations for this phantom
were randomly sampled from the uniform distribution on S2. We computed
the common line between each pair of projections, that is, the Fourier rays
Λk1,l1 and Λk2,l2, such that Λk1,l1 is closest to Λk2,l2 . Figure 5.3 shows the
dissimilarity between Λk1,l1 and Λk2,l2 for each pair of images k1 and k2, sorted
from the smallest (most similar) to the largest (most different). Such a plot
allows to pick the threshold for filtering the GCAR matrix, as described in
Section 4.
Figure 5.4a shows the spectrum of the operator A. Figure 5.4b presents
the orientation estimation error. The estimation error for each orientation
is defined as the angle (in radians) between the true orientation and the
estimated one. Finally, Fig. 5.1b shows the reconstructed phantom. We
can see that the reconstructed phantom is related to the original phantom
23
(a) Original (b) Reconstructed
Figure 5.1: Original and reconstructed analytic phantoms.
through an orthogonal transformation, which follows from the multiplicity of
the three-dimensional eigenspace.
In Figs. 5.5–5.8 we present the results for the E. coli ribosome density
map. Figure 5.5a shows the reference three-dimensional density map of the
E. coli ribosome. Several of its projections are given in Fig. 5.6. The orienta-
tions in which the projections of this phantom were computed are presented
in Fig. 5.7. The spectrum of the corresponding operatorA is given in Fig. 5.8.
Note that the multiplicities in this spectrum are different than the multiplic-
ities of the spectrum in Fig. 5.4a. The second eigenspace in Fig. 5.8 is of
dimension three as the coordinates x, y, and z are always exact eigenvectors,
as shown in Section 3.3. However, as opposed to the spectrum in Fig. 5.4a,
the next eigenspace is not of dimension five. Due to the distribution of the
projection orientations, shown in Fig. 5.7, the resulting GCAR operator does
not commute with rotations at all. Hence, the arguments in Section 3.4 do
not hold and the spectrum in Fig. 5.8 is not the spectrum of the spherical
harmonics. Finally, two views of the reconstructed density map are given in
Figs. 5.5b and 5.5c.
24
Figure 5.2: Sample of eight projection images of the analytic phantom.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010
−7
10−6
10−5
10−4
10−3
10−2
Figure 5.3: Semi-log plot of the dissimilarities between each pair of commonlines.
25
0 1 2 3 4 5 6 7 8 9 10 11 12 13 140
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
(a) Spectrum of the GCAR operator|1 − λi|
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
200
400
600
800
1000
1200
1400
1600
1800
(b) Error histogram
Figure 5.4: Spectrum of the GCAR operator for the analytic phantom andthe estimation error histogram.
(a) Original
(b) Reconstructed view 1 (c) Reconstructed view 2
Figure 5.5: Original and reconstructed E. coli density maps.
26
Figure 5.6: Sample of eight projection images of the E. coli ribosome densitymap.
−1
−0.5
0
0.5
1−1
−0.50
0.51
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5.7: Orientations on S2 used to generate the E. coli projections.
27
0 1 2 3 4 5 6 7 8 9 10 11 12 13 140
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Figure 5.8: Spectrum of the GCAR operator (|1 − λi|) for the E. coli densitymap.
6 Extensions
6.1 Center determination
The Fourier projection-slice theorem in Section 2.2 requires that for any
projection P taken in the direction θ, the center of the imaged object is
projected into the center of the projection image P . Such a center, however,
has no physical meaning, and any point in the three-dimensional object space
can be chosen as the center, by simply putting the origin of the coordinate
system at that point. Equation 3 states that this point should be projected
into the origin of the coordinate system in all projections.
In practice, each projection image is segmented from a much larger micro-
graph, containing many projection images, by roughly estimating the bound-
ing box of the imaged molecule’s copy. The projections obtained by such a
segmentation procedure would not satisfy Eq. 3 simultaneously, because their
centers are inconsistent. This means, for example, that defining the center
of each projection as its center of mass would lead to inconsistencies.
As a consequence, the input to the cryo-EM problem is a set of projections
Qg1, . . . , QgK , where each projection Qgk contains some unknown shift with
respect to its unshifted version Pgk to which the common line property of
Eq. 3 can be applied; that is, Qgk(xk, yk) = Pgk(x
k + ∆xk, yk + ∆yk), where
28
∆xk and ∆yk are the unknown shifts.
The 2D Fourier transform (see also Eq. 4) is given
f̂(ωx, ωy) =1
2π2
∫∫ ∞
−∞
f(x, y)e−ı(xωx+yωy)dx dy.
By the Fourier shift theorem, if g(x, y) = f(x + ∆x, y + ∆y) for some fixed
shifts ∆x and ∆y, then,
ĝ(ωx, ωy) = f̂(ωx, ωy)eı(∆xωx+∆yωy). (19)
Let Pg1 and Pg2 be two unshifted projections (centered with respect to the
center of the underlying three dimensional object). We assume that Pg1 uses
the coordinate system (x1, y1), and Pg2 uses the coordinate system (x2, y2).
Let
Qg1(x1, y1) = Pg1(x
1 + ∆x1, y1 + ∆y1),
Qg2(x2, y2) = Pg2(x
2 + ∆x2, y2 + ∆y2)
be the translated versions of Pg1 and Pg2, shifted by (∆x1,∆y1) and (∆x2,∆y2),
respectively. The Fourier shift theorem (Eq. 19) implies
Q̂g1(ω1x, ω
1y) = P̂g1(ω
1x, ω
1y)e
ı(∆x1ω1x+∆y1ω1y),
Q̂g2(ω2x, ω
2y) = P̂g2(ω
2x, ω
2y)e
ı(∆x2ω2x+∆y2ω2y).
(20)
Suppose that the common line of the projections P̂g1 and P̂g2 is (r cos θ1, r sin θ1)
in P̂g1 and (r cos θ2, r sin θ2) in P̂g2, with θ
1 and θ2 measured from the wx-axis
in P̂g1 and P̂g2, respectively. Along the common line
P̂g1(r cos θ1, r sin θ1) = P̂g2(r cos θ
2, r sin θ2), (21)
29
and so,
Q̂g1(r cos θ1, r sin θ1)e−ır(∆x
1 cos θ1+∆y1 sin θ1)
= Q̂g2(r cos θ2, r sin θ2)e−ır(∆x
2 cos θ2+∆y2 sin θ2),
from which we get
1
rarg
Q̂g1(r cos θ1, r sin θ1)
Q̂g2(r cos θ2, r sin θ2)
= µg1,g2 (22)
with
µg1,g2 = ∆x1 cos θ1 + ∆y1 sin θ1 − ∆x2 cos θ2 − ∆y2 sin θ2. (23)
Given K projection images, there are 2K unknowns(
∆xk,∆yk)
and(
K2
)
equations of the form of Eq. 22. Thus we form the(
K2
)
× 2K matrix system
of linear equations given by Eq. 22, and solve it using least squares. Note
that this linear system is very sparse as each row contains only four nonzero
elements. The resulting matrix has a null-space of dimension three, which
reflects the fact that arbitrarily moving the origin of the object space R3
induces another set of consistent translations(
∆xk,∆yk)
in the projections,
which also satisfy Eq. 22. As in the case of constructing the matrix W
in Section 3.1, we need not use all(
K2
)
equations, but only equations that
correspond to highly similar common lines. We can also further filter the
system by choosing only equations that correspond to pairs for which the
left hand side in Eq. 22 in nearly constant for various values of r. Although
in theory the left hand side of Eq. 22 should be constant for all r, this is not
the case in practice due to discretization, noise, and measurement errors.
To form the translation estimation equations described above, we need to
detect common lines between pairs of projections in the presence of unknown
relative shifts. As a result of Eq. 21
|P̂g1(r cos θ1, r sin θ1)| = |P̂g2(r cos θ
2, r sin θ2)|,
30
and from Eq. 20 we get
|Q̂g1(r cos θ1, r sin θ1)| = |Q̂g2(r cos θ
2, r sin θ2)|.
Hence, to detect common lines between projections that were shifted by some
unknown shift, we take the polar Fourier transform of each projection, and
find common lines between the magnitude of the Fourier rays.
It may be that two Fourier rays have the same absolute values, although
these are not the common line between the two projections. To overcome this
problem we find not only the closest pair of rays in |Q̂g1 | and |Q̂g2|, but several
such pairs. For each pair we assume it is the common line and estimate the
phase factor µg1,g2 in Eq. 23 using Eq. 22. Thus we can compute the similarly
between the rays P̂g1(r cos θ1, r sin θ1) and P̂g2(r cos θ
2, r sin θ2) by computing
the correlation between Q̂g1(r cos θ1, r sin θ1)e−ırµg1,g2 and Q̂g2(r cos θ
2, r sin θ2),
and choosing the pair θ1 and θ2 that brings this correlation to maximum. We
then use these θ1 and θ2 to form a common line equation of the form of Eq. 22.
This results, as explained above, in(
K2
)
equations for the 2K unknown shifts,
which we solve using least-squares.
The performance of the center determination algorithm is demonstrated
in Fig. 6.1. We used K = 100 projections of the analytic phantom (Fig. 5.1a),
each of size 97× 97 pixels. An odd size was chosen for the projection images
to avoid inherent half-pixel shifts associated with even sampling sizes. Each
projection was randomly shifted in the x and y directions by some random
shift of up ±20 pixels in each direction. For each projection we computed L =
300 radial Fourier lines, each with n = 100 samples in the radial direction.
Figure 6.1a shows the 10 smallest singular values of the center estimation
system, given by Eq. 22. The null space of dimension three is apparent.
Figure 6.1b shows the shifts estimation error. Each bar corresponds to the
absolute estimation error (in pixels) in one of the K = 100 projections.
31
1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
(a) Smallest singular values of the(
K
2
)
×2K sparse matrix obtained from Eq. 22
10 20 30 40 50 60 70 80 90 1000
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
(b) Absolute estimation error
Figure 6.1: Performance of the center estimation algorithm.
(a) Molecule of type 1 (b) Molecule of type 2
Figure 6.2: Two types of molecules (3D rendering of the analytic phantoms).
6.2 Heterogeneity problem
Suppose our data set contains projections of two different types of molecules.
In this section we describe how to use the spectral properties of the operator
A defined in Section 3.2 to discriminate between the two different molecules.
We will accompany the explanation with an example using two analytic phan-
toms, depicted in Figs. 6.2a and 6.2b. The example shows only two types
of molecules, but essentially the same algorithm can be used to handle more
than two types.
Suppose we have a mix of K1 projections from the first molecule and K2
32
projections from the second molecule. Assume for the sake of presentation
that the first K1 projections belong to type 1, and the next K2 projections
belong to type 2. Suppose that we find the common line between any pair of
images. For each of the(
K1+K22
)
pairs of images, we obtain the lines Λk1,l1
and Λk2,l2, which are supposed to be a common line Λk1,l1 = Λk2,l2 . We
measure the dissimilarity between Λk1,l1 and Λk2,l2 and obtain a dissimilarity
coefficient disk1,k2 corresponding to the pair of projections (k1, k2). We expect
that on average, the dissimilarity coefficient for two projections that belong
to the same type of molecule will be smaller (more similar) than for two
projections of different types. Thus, if we sort the dissimilarity coefficients
dissk1,k2, k1, k2 = 1, . . . , K1 + K2 from smallest to largest, we expect that
the first part of the sorted array will correspond to dissimilarities between
projections from the same type of molecule, and the second part of the sorted
array, which corresponds to larger dissimilarity coefficients, will correspond
to common lines between projections of different types.
To demonstrate this point, we generated a mix that contains K1 = 100
random projections of the molecule of type 1 (Fig. 6.2a) and K2 = 100
random projections of the molecule of type 2 (Fig. 6.2b). The random ori-
entations used for each type of molecule are independent. Each projection
is of size 96 × 96 pixels. To determine common lines, we computed L = 300
radial Fourier line in each projection (angular resolution), and used n = 100
samples on each Fourier line (radial resolution). We computed the dissimi-
larity coefficient for each pair of images. The sorted dissimilarity coefficients
are shown in Fig. 6.3. As expected, the first part of the sorted list contains
small dissimilarity coefficients (roughly half of the list since K1 = K2 = 100).
The second part of the sorted list contains significantly larger values.
If we filter the GCAR matrix W , defined in Section 3.1, as descried in
Section 4, by retaining only common lines that correspond to the first part of
the graph in Fig. 6.3, then the resulting matrix of size (K1+K2)L×(K1+K2)L
is a block matrix with two non-interacting blocks. Hence, the multiplicities
33
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
10−7
10−6
10−5
10−4
10−3
10−2
Figure 6.3: Dissimilarity coefficient sorted from smallest to largest.
of the eigenvalues will be doubled compared to the case of a single type of
molecule. The spectrum of A for the given heterogeneity example is given
in Fig. 6.4.
The first eigenspace (corresponding to eigenvalue 1, or |1 − λ1| = 0 in
Fig. 6.4) is of dimension two, and the second eigenspace is of dimension
six. The first eigenvector, which in the single molecule case is the all-ones
vector, now becomes piecewise constant. In fact, by orthogonalizing the first
eigenspace of dimension two relative to the all-ones vector, we get a piecewise
constant vector w ∈ R(K1+K2)L such that wj > 0 if Fourier line j belongs to
a projection of the first type, and wj < 0 otherwise. The first eigenvector of
the mixed GCAR matrix is presented in Fig. 6.5. Figure 6.5 corresponds to
the case where the K1 +K2 projections are randomly shuffled. Only 15000
out of the 60000 entries of the first eigenvector are shown.
We can therefore partition all Fourier lines according to the type of their
underlying molecule, and thus classify the projections according to their type.
Once partitioned into two classes, the orientations in each class can be deter-
mined separately. Figures 6.6a and 6.6b show the sorted dissimilarity coeffi-
34
0 1 2 3 4 5 6 7 8 90
0.005
0.01
0.015
0.02
0.025
Figure 6.4: Spectrum of the filtered GCAR matrix (|1 − λi|) constructedfrom a mix of two types of molecules.
0 5000 10000 15000−5
−4
−3
−2
−1
0
1
2
3
4
5x 10
−3
Figure 6.5: First few values of the first eigenvector of the mixed GCARmatrix (after orthogonalizing to the all-ones vector).
35
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000
0.2
0.4
0.6
0.8
1
1.2x 10
−4
(a) Molecule of type 1
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000
1
x 10−4
(b) Molecule of type 2
Figure 6.6: Dissimilarity coefficients for each type of molecule.
0 1 2 3 4 5 6 7 8 90
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
(a) Molecule of type 1
0 1 2 3 4 5 6 7 8 90
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
(b) Molecule of type 2
Figure 6.7: Spectrum of the GCAR matrix for each type of molecule.
cient for each type of molecule after classification. The plots in Figs. 6.6a and
6.6b are then used to separately filter the two GCAR matrices corresponding
to the individual molecules.
Figures 6.7a and 6.7b show the spectrum of the GCAR matrix of each
class. The spectrum for each class is as predicted: the first eigenvalue is 1, and
the next subspace is of dimension three. We then estimate the orientations of
the projections in each class separately. Histograms of the error estimation
are given in Figs. 6.8a and 6.8b.
The advantage of the above procedure is that it uses all the data simul-
36
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450
500
1000
1500
2000
2500
3000
estimation error
(a) Molecule of type 1
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
1000
2000
3000
4000
5000
6000
estimation error
(b) Molecule of type 2
Figure 6.8: Histogram of the orientation estimation error for each type ofmolecule.
taneously to separate the classes, as opposed to other methods, which try to
classify each projection separately.
6.3 Particle selection
The data acquisition process in cryo-electron microscopy generates many
projection images which are corrupted by noise, improper segmentation, and
overlapping particles, to name but a few. It is desirable to be able to remove
such particles from the data set before any reconstruction process takes place.
The matrixW , defined in Section 3.1, can be used to identify such corrupted
projections. Once constructing the matrix W , we filter it as described in
Section 4. At this point we inspect how many projections intersect a given
projection such that the common line between the two projections is highly
similar. In theory, in a noiseless setting, any two projections should have a
common line. That is, each projection image should have a total of O(K)
interactions with other projections. When a projection is corrupted by noise
or is otherwise inconsistent with the rest of the projections, there would
be only a few such intersections. This would suggest that this projection
is inconsistent with the rest of the data set and should be rejected. This
37
Figure 6.9: Corrupted projections.
again demonstrates the principle that all the data at once should be used to
determine which projections are to be used for reconstruction: a projection
is “good” if many other projections “say” it is good, by having a common
line with it.
To demonstrate this idea we generated K = 100 projections of the an-
alytic phantom in Fig. 5.1a, and corrupted 10 projections by a rectangle of
Gaussian noise of size up to 10×10 pixels. See Fig. 6.9 for several of these cor-
rupted projections. We then constructed the matrix W as described above,
filtered it, and rejected projections that intersect with less than 10 percent
of the other projections. Figure 7.1 shows the number of intersections for
each projection. The first 20 projections are the corrupted ones. It is clear
that by removing projections with a few intersections we retain the “good”
projections.
7 Summary and Future Work
In this technical report we introduced a new methodology for the three di-
mensional cryo-EM structure determination. Our GCAR algorithm incorpo-
rates the Fourier-projection slice theorem into a novel construction of a graph,
followed by an efficient calculation of a few eigenvectors of the normalized
sparse adjacency matrix. The resulting eigenvectors reveal the projection ori-
entations in a globally consistent manner. We demonstrated the success of
the method when applied to artificially produced projection images, as well
38
as its applicability to the image centering and the heterogeneity problems.
The reader must be asking herself if this method would also be successful
in practice, when faced with real noisy images, rather than artificially pro-
duced clean images. We have good reasons to believe that this would be the
case, but at this point of time, we prefer leaving speculation aside. One has
to keep in mind that this technical report summarizes a work which is still in
progress, and we hope that soon enough we will be able to provide a definite
answer.
Not included in this technical report are preliminary results regarding
the behavior and success of the GCAR algorithm when the input images are
corrupted by white Gaussian noise. Detection of common lines in the pres-
ence of noise is much more difficult: out of all detected common lines, only
a small percentage are actually true common lines. Although the resulting
embedding found by the GCAR algorithm is distorted and cannot be used
to reveal the orientations, it can be iteratively improved until convergence
to a globally consistent embedding is obtained. In each iteration, we ignore
common lines that do not agree with the previously obtained embedding.
This iterative procedure, which resembles well known procedures in robust
estimation, such as the iterative weighted least squares procedure, cleans up
the noisy graph and enables successful reconstructions.
Clearly, denoising of either projection images or radial Fourier lines is a
major theme of our future work. We plan to apply both classical and modern
denoising methods from image and signal processing. There are two main
approaches for denoising the projection images. In the first approach, each
projection image will be denoised separately, and the denoised images will
be compared to find the common lines. In the second approach, we will
try to denoise many images at once, since similar features should appear
in different images. A complete discussion of denoising methods and their
practical success will be the subject of a future report.
39
10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
Figure 7.1: Number of intersections of each projection with other projectionsafter filtering the GCAR matrix.
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